#7080858

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,29,24,30,22,28)(19,31,33,27,34,43,21,25)(20,35,38,46,39,26,41,45)(32,48,37,42,36,47,40,44);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,24,34)(18,25,30,27)(20,36,39,32)(21,22,33,23)(26,44,35,42)(28,31,29,43)(37,41,40,38)(45,47,46,48);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20,24,39)(18,26,30,35)(19,32,34,36)(21,40,33,37)(22,41,23,38)(25,42,27,44)(28,46,29,45)(31,47,43,48);(3,8)(5,7)(11,16)(13,15)(17,21,23,19,24,33,22,34)(18,27,29,43,30,25,28,31)(20,37,38,36,39,40,41,32)(26,42,45,47,35,44,46,48);(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,21,34,33)(20,38,39,41)(25,43,27,31)(26,45,35,46)(32,37,36,40)(42,47,44,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=h,d^2=fgi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6988054);